Dependence of the blow-up time with respect - Universidad de ...

DEPENDENCE OF THE BLOW-UP TIME 9
As another example if Ω is a cube, Ω = (0, 1) n , we can use a semidiscrete
finite differences method to approximate the solution u(x, t) obtaining
an ODE system of the form (2.3).
We require to the general scheme that we introduce here to be consistent.
A precise definition of consistency is given below
Definition 2.1. We say that the scheme (2.3) is consistent if for any
solution u of (2.1) holds
N
(2.4) mkut(xk, t) = − akju(xj, t) + mku p (xk, t) + ρk(h, t),
j=1
and there exists a function ρ : R+ → R+ depending only on h and a
universal constant θ such that
max
ρk,h(t)
ρ(h)
k ≤ , for every t ∈ (0, Th),
(T − t) θ
mk
with ρ(h) → 0 if h → 0.
We want to remark that this consistency hypothesis is valid in the
two examples cited above with ρ(h) = Ch 2 . The power C(T − t) −θ is
a bound for the spatial derivatives of u.
Using ideas from [GR] and [GQR] it can be proved that the method
converges uniformly in sets of the form Ω × [0, T − τ]. Moreover, using
the energy functional
Φh(U(t0)) ≡ 1
2 〈A1/2U(t0); A 1/2 N+1
U(t0)〉 −
i=1
mii
((U(t0))i) p+1
,
p + 1
one can check that, given any initial data u0 such that the continuous
solution u blows up, then the numerical approximation uh also blows
up in finite time, Th, for every h small enough. Since our interest here
is to analyze the convergence rate of |T − Th| we refer to those papers
([GR] and [GQR]) for the details.
Before involving us in the main result of this part we will prove some
lemmas. We will need the following definition,
Definition 2.2. We say that U is a supersolution of (2.2) if
MU ′ ≥ −AU + MU p .
We say that U is a subsolution of (2.2) if
MU ′ ≤ −AU + MU p .